Section 5–7 Short distances

(Molecular sizes / Nuclear sizes / Uncertainty principle)

In this section, the three interesting points discussed are molecular sizes, nuclear sizes, and the uncertainty principle.

1. Molecular sizes:

“…Electron micrograph of some virus molecules. The ‘large’ sphere is for calibration and is known to have a diameter of 2 × 10⁻⁷ meter (Feynman et al., 1963, section 5.7 Short distances).”

In an optical microscope, we can measure a small object by using an eyepiece reticule which is a small piece of glass insert with a ruler scale inscribed in it. This method of measurement is limited by the wavelength of visible light (about 5 × 10⁻⁷ meter) that “see” microscopic objects. For smaller objects such as virus molecules, we need to use electron microscopes instead of optical microscopes. In short, an electron microscope utilizes electromagnetic lenses to focus electrons into a very thin beam as compared to an optical microscope that utilizes glass lenses to focus a light beam.

We can continue to measure sizes of microscopic objects to smaller scales by selecting electromagnetic waves that have shorter wavelengths. For x-rays, we need to first determine its very short wavelength by observing the diffraction angles from a plane ruled grating. Then, from a measurement of the pattern of the scattering of the x rays from a crystal, we can determine the atomic spacings that have a dimension of about 10⁻¹⁰ m. It is worth mentioning that the Nobel Prize in Physics 1914 was awarded to Max von Laue for his discovery of the diffraction of X-rays by crystals and the Nobel Prize in Physics 1915 was awarded to Sir William Henry Bragg and his son William Lawrence Bragg for their analysis of crystal structure by using X-rays.

2. Nuclear sizes:

“…we find that the radii of the nuclei are from about 1 to 6 times 10⁻¹⁵ meter. The length unit 10⁻¹⁵ meter is called the fermi, in honor of Enrico Fermi (Feynman et al., 1963, section 5.7 Short distances).”

Dr. Sands explains a measurement of nuclear sizes by passing high energy particles through a thin slab of material and by observing the number of particles which come out. This method is based on the chance that some small particles will hit the nuclei in a trip. Suppose in an area A of the slab that has N atoms, the fraction of the area “covered” by the nuclei is about Nσ/A in which σ is the apparent area of a nucleus (or the effective cross section). If the number of particles of a beam which arrive at the slab is n₁ and the number of particles which come out from the other end is n₂, then the fraction which is blocked is (n₁ − n₂)/n₁, which is possibly equal to the fraction of the area that is covered or blocked. Thus, we can deduce the radius of the nucleus from the equation Nσ/A = (n₁ − n₂)/n₁ by rewriting it as πr² = σ = (A/N)(n₁ − n₂)/n₁.

Note: This is a crude measurement of nuclear sizes. Dr. Sands elaborates the concept of probability that a particle will experience a collision in a slab in section 6.1 Chance and likelihood.

In his Nobel lecture titled The electron-scattering method and its application to the structure of nuclei and nucleons, Hofstadter (1961) mentions that “in the year 1919 the first vague ideas concerning the sizes of nuclei were worked out. By studying the deviations from Coulomb scattering of alpha particles Rutherford showed that a nuclear radius was of the order of 10⁵ times smaller than an atomic radius (p. 560).” He adds that “[w]e have used the method of high-energy electron scattering. In essence, the method is similar to the Rutherford scattering technique, but in the case of electrons it is presently believed that only a simple and well-understood interaction - the electromagnetic or Coulomb interaction - is involved between the incident electron and the nucleus investigated (p. 561).” Mathematically, quantum electrodynamics and Dirac theory can be used to calculate a differential elastic scattering cross section.

3. Uncertainty principle:

“…Perfectly precise measurements of distances or times are not permitted by the laws of nature (Feynman et al., 1963, section 5.7 Short distances).”

According to Dr. Sands, perfectly precise measurements of distance and time are not permitted by Heisenberg’s uncertainty principle. Interestingly, his explanation includes a perspective of the special theory of relativity such that measurements of distance and time are dependent on an observer’s frame of reference. He elaborates that the errors in a measurement of the position of an object must be minimally as large as Δx ≥ ℏ/2Δp where ℏ is the reduced Planck constant and Δp is the error in our knowledge of the momentum of the object whose position we are measuring. However, the term “error” should be avoided because it has a connotation of mistake. Thus, physicists prefer the word uncertainty which may mean a lack of knowledge in determining these physical quantities or experimental inaccuracy instead of an error.

Feynman’s explanation of uncertainty principle is related to the Young double slit experiment. In Feynman’s words, “If you make the measurement on any object, and you can determine the x-component of its momentum with an uncertainty Δp, you cannot, at the same time, know its x-position more accurately than Δx ≥ ℏ/2Δp. The uncertainties in the position and momentum at any instant must have their product greater than or equal to half the reduced Planck constant. This is a special case of the uncertainty principle that was stated above more generally. The more general statement was that one cannot design equipment in any way to determine which of two alternatives is taken, without, at the same time, destroying the pattern of interference (Feynman et al., 1963, Section 37–8 The uncertainty principle).”

Questions for discussion:

1. How do physicists measure the size of molecules?

2. How do physicists measure the size of nuclei?

3. Does the uncertainty principle simply mean a complete precise measurement of the location of an object will result in a complete uncertainty in its momentum?

The moral of the lesson: physicists are able to measure the sizes of molecules and nuclei, however, it is debatable whether it is possible to measure smaller dimensions as stipulated by the uncertainty principle.

References:

1. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.

2. Hofstadter, R. (1961). The Electron Scattering Method and its Application to the Structure of Nuclei and Nucleons. In Nobel Lectures in Physics 1942 – 1962. Singapore: World Scientific.